动载荷作用下无限长FGM梁在分数阶黏弹性地基上的精确解

Exact Solution for an Infinite FGM Beam on Frational Order Viscoelastic Foundation Subjected to Dynamic Loads

基于分数阶微分Zener型黏弹性地基模型,建立动载荷作用下无限长FGM梁在分数阶黏弹性地基上的运动控制微分方程。利用傅里叶和拉普拉斯变换将控制微分方程简化为代数方程。首先在频率域内得到解答,然后利用傅里叶和拉普拉斯逆变换以及卷积定理将解答再转换回时间域内,得到黏弹性地基上FGM梁的挠度、速度、加速度、弯矩和剪力响应的精确解。最后,计算了冲击荷载作用下弹性地基FGM梁的动态响应,给出了x=0处梁的垂直速度和弯矩的响应曲线,其形状特征和均匀材料梁相同,且材料梯度指标p对结果的影响较小。

Based on the fractional Zener model,the governing differential equations of motion for an infinite FGM beam resting on a frational order viscoelastic foundation subjected to dynamic loads are formulated. The Fourier and Laplace transforms are utilized to simplify the governing differential equation of the beam to an algebraic equation,so that the solution can be conveniently obtained in the frequency domain. The inverse Laplace transform,inverse Fourier transform,and convolution theorem are employed to convert the solution into the time domain and the exact solution for responses of deflection,velocity,acceleration,bending moment,and shearing force of the FGM beam are obtained. Finally,the dynamics responses of the FGM beams on elastic foundation subjected to shock loads are calculated and vertical velocity and bending moment responses curves at the position of x = 0 are shown. The results show that shape features of the curves are similar to those of homogeneous material beams and material gradient index p has few effect on the results.